function [J, grad] = linearRegCostFunction(X, y, theta, lambda)
%LINEARREGCOSTFUNCTION Compute cost and gradient for regularized linear 
%regression with multiple variables
%   [J, grad] = LINEARREGCOSTFUNCTION(X, y, theta, lambda) computes the 
%   cost of using theta as the parameter for linear regression to fit the 
%   data points in X and y. Returns the cost in J and the gradient in grad

% Initialize some useful values
m = length(y); % number of training examples

% You need to return the following variables correctly 
J = 0;
grad = zeros(size(theta));

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost and gradient of regularized linear 
%               regression for a particular choice of theta.
%
%               You should set J to the cost and grad to the gradient.
%

h = X * theta;
err = h - y;
J = sum( err .^ 2 ) / ( 2 * m );

addtheta = sum( theta( 2 : end ) .^ 2 ) * lambda / ( 2 * m );
J = J + addtheta;

grad = ( X' * err ) / m;
grad_theta = theta;      
grad_theta( 1 ) = 0;
add_grad_theta = ( grad_theta * lambda ) / m;
grad = grad + add_grad_theta;

% =========================================================================

grad = grad(:);

end
